We present a simple algorithm that successfully re-constructs a sine wave, sampled vastly below the Nyquist rate, but with sampling time intervals having small random perturbations. We show how the fact that it works is just common sense, but then go on to discuss how the procedure relates to Compressed Sensing. It is not exactly Compressed Sensing as traditionally stated because the sampling transformation is not linear.  Some published results do exist that cover non-linear sampling transformations, but we would like a better understanding as to what extent the relevant CS properties (of reconstruction up to probability) are known in certain relatively simple but non-linear cases that could be relevant to industrial applications.

We show that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on smoothness in a neighborhood of a solution. Given that the problem is convex, our method will converge even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including matrix factorization and training of ResNet-18.

Many manufacturing processes require the use of robots to transport parts around a factory line. Some parts, which are very thin (e.g. car doors)
are prone to elastic deformations as they are moved around by a robot. These should be avoided at all cost. A problem that was recently raised by
F.E.E. (Fleischmann Elektrotech Engineering) at the ESGI 158 study group in Barcelona was to ascertain how to determine the stresses of a piece when
undergoing a prescribed motion by a robot. We present a simple model using Kirschoff-Love theory of flat plates and how this can be adapted. We
outline how the solutions of the model can then be used to determine the stresses. 

Many real networks feature the property of nestedness, i.e. the neighbours of nodes with a few connections are hierarchically nested within the neighbours of nodes with more connections. Despite the abstract simplicity of this notion, different mathematical definitions of nestedness have been proposed, sometimes giving contrasting results. Moreover, there is an ongoing debate on the statistical significance of nestedness, since even random networks where the number of connections (degree) of each node is fixed to its empirical value are typically as nested as real-world ones. In this talk we show unexpected effects due to the recent finding that random networks where the degrees are enforced as hard constraints (microcanonical ensembles) are thermodynamically different from random networks where the degrees are enforced as soft constraints (canonical ensembles). We show that if the real network is perfectly nested, then the two ensembles are trivially equivalent and the observed nestedness, independently of its definition, is indeed an unavoidable consequence of the empirical degrees. On the other hand, if the real network is not perfectly nested, then the two ensembles are not equivalent and alternative definitions of nestedness can be even positively correlated in the canonical ensemble and negatively correlated in the microcanonical one. This result disentangles distinct notions of nestedness captured by different metrics and highlights the importance of making a principled choice between hard and soft constraints in null models of ecological networks.

[1] Bruno, M., Saracco, F., Garlaschelli, D., Tessone, C. J., & Caldarelli, G. (2020). Nested mess: thermodynamics disentangles conflicting notions of nestedness in ecological networks. arXiv preprint arXiv:2001.11805.
 

We consider random graph models where graphs are generated by connecting not only pairs of nodes by edges but also larger subsets of
nodes by copies of small atomic subgraphs of arbitrary topology. More specifically we consider canonical and microcanonical ensembles
corresponding to constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such
models. We also introduce a procedure that enables the distributions of multiple atomic subgraphs to be combined resulting in more coarse
grained models. As a result we obtain a general class of models that can be parametrized in terms of basic building blocks and their
distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs (Karrer & Newman PRE 2010, Bollobas et al. RSA 2011), random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy expressions for all these models can be derived from a single expression that is characterized by the symmetry groups of their atomic subgraphs.

We will discuss some problems around independence of l in compatible systems of Galois representations, mostly focusing on the independence of l of algebraic monodromy groups. We will explain how these problems fit into the context of the Langlands program, and present results both in characteristic zero and in positive characteristic settings.